Tuesday, November 30, 2010

More on Zeno

I asked earlier how the four premisses of Zeno's argument given in the IEP imply the conclusion that Achilles never reaches the tortoise.

Clearly there are other assumptions that have to be made. There is the ‘and so on’ of premiss 4. But how does that work? Suppose Achilles aims at the exact spot Y where he is going to overtake the tortoise. Clearly when he has reached that spot, he will have reached the tortoise. If he reaches any spot X before that, he will have not reached the tortoise. So all the ‘argument’ appears to be saying, it seems, is that if we take any point X before Y, then there is some distance to go. And if we take any spot X’ between Y and X, there is still some distance to be ‘and so on’. But this ‘and so on’ doesn’t prove anything. It proves simply (or rather, it assumes that) we can take any distance whatever, and cut it somewhere. How does that prove ‘Achilles will never reach the tortoise’?

Graham Priest has a slightly different version of the argument. He says (I paraphrase) In order to get from a to b, you must first get to each point between a and b. But there are infinitely many points between a and b. Hidden premiss: to get to something = to do something. Therefore to get from a to b in a finite time, you must do infinitely many things. But you can’t do infinitely many things in a finite time. Therefore etc. But there is much to challenge there. Is the hidden premiss correct? Is ‘getting to’ a point the same as ‘doing something’? Can we actually ‘get to’ a mathematical point? How? We can cross such a point, of course. But then the argument amounts to this: there are infinitely many collections of finite distances between a and b, and we can traverse any such collection in a finite time. Indeed, clearly we can, for the total length of any such collection will be the length between a and b.

Aristotle mentions the argument several times in the Physics, arguing that we must distinguish the ‘actual’ from the ‘potential’ infinite, but this distinction is not very clear. I don't have the book to hand, so will post something later.


David Brightly said...

Brandon appears to take the argument to show that if A is to pass the T then he must reach or pass an infinite number of points. Together with the assumption that A cannot perform infinitely many acts in a finite time we seem to have a contradiction. I think you are right to ask the question: Is ‘getting to’ [or passing] a point the same as ‘doing something’? The problem can be formulated without any of the sophisticated mathematics of the continuum as suggested by the IEP article and implied by Brandon ("the extraordinarily complex) Standard Solution"). We can take all the distances and times as rational. We can attack this in three ways:
1) We can say that the divisions are purely conceptual: 1 cake = 1/2 cake + 1/4 cake + 1/8 cake + 1/16 cake + ... without actually cutting the cake. This perhaps just comes down to the SS.
2) We can ask in what sense these actions are additive.
3) We can say that the concept of 'number of actions in getting from A to B' is not well defined. For (a) we can split the passage from A to B into any number of sub-passages we like. Or (b) we can say that the man sitting in the train gets from A to B by doing nothing, ie, Galilean inertia, Newton's first law.

That getting from A to B is an 'action' is exactly what the paradox is telling us, in Brandon's phrase, 'we can't accept'.

Edward Ockham said...

Hello David - welcome back :)

David Brightly said...

Thank you. A further thought. Here is your version of the argument at Brandon's:

(1) If at every point Achilles reaches there are points that still have to be traversed to reach the tortoise, Achilles always has some distance to go to catch up with the tortoise

(2) At every point Achilles reaches there are points that still have to be traversed to reach the tortoise

(3) Therefore Achilles always has some distance to go to catch up with the tortoise (i.e. will never catch the tortoise).

You ask for justification for (2). We can cast doubt on (1) as well. The antecedent contains a quantification ('every point') over the sequence of spatial points that the argument generates. The consequent quantifies over time ('always'). The conditional is patently false.

Edward Ockham said...

Very true. I have another idea I will post tomorrow. Hope you are managing in this weather.